The orthic triangle: excentres

Another property, very similar to the first, is that A, B and C are the excentres of the orthic triangle.

orthic triangle 4

I will deal with the acute case here – in the obtuse case, the result is slightly different, as one excircle trades places with the incircle of DEF.

This follows on easily from the first result: using the fact that interior angle bisectors and exterior angle bisectors are perpendicular, the sides of ABC must form the exterior angle bisectors of DEF because they are perpendicular to the altitudes, which, as we established, are the interior angle bisectors of DEF.

Hence their intersections – ie the vertices – are the centres of the excircles of DEF.

In other words, the incentre of a triangle is the orthocentre of its excentral triangle.

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